math

February Above & Beyond

Due Friday, Feb. 27, 2015

Name:

Directions: Complete the following problems. Show your work and justify your answers.

1. Fibonacci Sequence

The Fibonacci sequence is a famous list of numbers named after the Italian merchant (c. 1170 CE-c. 1250 CE) who popularized the Arabic numerals (0-9) that we use today. The sequence appeared before Fibonacci in ancient Indian writings.

The sequence starts with two 1’s, and then every other number is formed by adding the two preceding numbers. So the sequence begins

1, 1, 2, 3, 5, 8, . . .

and continues forever.

(a) Write the first ten numbers in the Fibonacci sequence (including the numbers above).

(b) Fill in the chart below:

Sum of first two Fibonacci numbers: Fourth Fibonacci number:

Sum of first three Fibonacci numbers: Fifth Fibonacci number:

Sum of first four Fibonacci numbers: Sixth Fibonacci number:

Sum of first five Fibonacci numbers: Seventh Fibonacci number:

(c) Look for a pattern in part (b). The numbers in the first column should relate to the numbers in the second column. Looking along each row, what pattern do you see?

(d) Use your discovery in part (c) to find the sum of the first 8 Fibonacci numbers (without adding them up). Explain how you got your answer.

The Fibonacci numbers are not just some weird mathematical pattern. They show up in nature: in the spiraling scales on pineapples and pinecones, the spiraling “florets” on the face of a sunflower, and the petals of daisies and roses! Why do Fibonacci numbers show up in nature like this? Often, it occurs because plants, through natural selection, have come up with the most efficient organization of their parts.

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2. The Golden Ratio

The ancient Greeks were geometers, which means they did essentially all of their math- ematics through geometry (they did not have an algebraic system like we have today). As such, they often viewed numbers in a geometric way (i.e., as lengths or areas of various shapes).

Suppose you have a wooden board that you want to cut into two pieces. The most aesthetically pleasing way to do this, according to the ancient Greeks, is to cut it according to the Golden Ratio: cut the board into two pieces, one longer and one shorter, so that the ratio of the longer to the shorter is equal to the ratio of the total board length to the longer. In symbols:

Golden Ratio = Total board length

Longer piece =

Longer piece

Shorter piece (1)

Let’s figure out the value of this Golden Ratio using algebra.

(a) You want to cut a board into the Golden ratio. To make our lives easy, let’s assume that we start with a wooden board that is 1 meter long. Let’s say the longer piece has length x. What is the length of the shorter piece in terms of x?

1︷ ︸︸ ︷ cut

short

piece

long

piece

x

Longer board length: x

Shorter board length:

(b) Now rewrite equation (1) in terms of x, using your answers to (a).

Total board length

Longer piece =

Longer piece

Shorter piece

1 = (2)

(c) Solve your equation from part (b). (You’ll probably have to use the quadratic formula at some point.) Show your work. Simplify your answers, but leave your answers in exact form.

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(d) [Calculator allowed] In part (c), you got two solutions. Approximate these so- lutions to three decimal places and plot them on the number line below.

−2 −1 0 1 2

You have two solutions, but only one makes sense in this context. Which solution makes sense? Why?

(e) Using parts (c) and (d), determine the length of the two pieces of the board below. In the second column, determine the exact length of each piece, using your answers from part (c). In the third column, approximate the lengths to three decimal places (using part (d) and a calculator).

Piece of board Exact length Approximate length

Longer piece

Shorter piece

(f) Now we can find the Golden ratio. From what we know above, the Golden ratio will be:

Golden Ratio = 1

Longer piece

i. Find the exact value of the Golden Ratio using the exact value from (e). Rationalize the denominator and simplify.

ii. Now approximate the Golden Ratio to three decimal places, using a calculator and the approximate value from (e)

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3. Interesting Connections Though it is not obvious, the Golden Ratio and the Fi- bonacci sequence are related. Let’s explore some of those connections.

(a) Fill in the chart below, with the help of a calculator. In the left column, write the ratio of consecutive Fibonacci numbers (with the larger number on top). In the right column, write the corresponding decimal approximation (round to three decimal places when appropriate). The chart has been started for you.

Ratio of consecutive Fibonacci numbers Decimal Approximation

1/1 1

2/1 2

3/2 1.5

5/3

8/5

(b) The ratios in part (a) are approaching some special number. Looking at the second column, can you guess what special number they are approaching?

(c) On a computer, navigate to wolframalpha.com. In the search field, enter exactly the text in each line below (before the colon). The program knows the value of the golden ratio and computes the value of your entry. Write the corresponding answers below.

(golden ratio^1 - (-1/golden ratio)^1)/sqrt(5) : _______________

(golden ratio^2 - (-1/golden ratio)^2)/sqrt(5) : _______________

(golden ratio^3 - (-1/golden ratio)^3)/sqrt(5) : _______________

(golden ratio^4 - (-1/golden ratio)^4)/sqrt(5) : _______________

(golden ratio^5 - (-1/golden ratio)^5)/sqrt(5) : _______________

(golden ratio^6 - (-1/golden ratio)^6)/sqrt(5) : _______________

(golden ratio^7 - (-1/golden ratio)^7)/sqrt(5) : _______________

(golden ratio^8 - (-1/golden ratio)^8)/sqrt(5) : _______________

(golden ratio^9 - (-1/golden ratio)^9)/sqrt(5) : _______________

(d) What is the name of the sequence of numbers that you get in part (c)?

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